Calculus: Intermediate



Overview

The tools of calculus are essential to our understanding of the universe, and as such they form the basis for the majority of mathematical models, from the spread of disease to the physics of glaciers.

The course begins with the familiar example of Ordinary Differential Equations (ODEs) and revisits some of the most-common methods of solution such as Integrating Factors, Homogeneous Functions, and Separation of Variables. We then use apply the tools of one-variable calculus to solve some problems related to optimisation.

Next, we introduce the concept of multi-variable functions through Partial Differentiation, Taylor's Theorem, and solving simple PDEs. We will also briefly touch on Vector Calculus in the form of the Gradient Vector and the Divergence.

The latter part of the course focuses on the specific example of the Heat Equation - one of the most fundamental PDEs and the gateway to the method of Fourier Series. 

We will end with a closer look at some applications of our newly discovered techniques to real-world problems, such as disease modelling and ice flow in a glacier, and (time-permitting) a short introduction to Integration and Jacobians.

This is an ‘intermediate’ FHEQ level 4 [https://en.wikipedia.org/wiki/National_qualifications_frameworks_in_the_United_Kingdom] course and therefore in order to get the most out of the teaching you should have some familiarity with Calculus as a pre-requisite. In particular, a knowledge of differentiation is a must. Taking the OUDCE 'Beginning Calculus' course would be ample preparation.

The overall structure of the course follows the Undergraduate Mathematics Syllabus at the University of Oxford. The topics covered each week are listed below.

Programme details

Courses starts: 28 April 2025

Week 1: Differential Equations

Week 2: Solving ODEs

Week 3: Optimisation

Week 4: Partial Differentiation

Week 5: Taylor’s Theorem and Critical Points

Week 6: Partial Differential Equations

Week 7: Heat Equation

Week 8: Fourier Series

Week 9: Applications

Week 10: Integrals

Certification

Credit Application Transfer Scheme (CATS) points 

To earn credit (CATS points) for your course you will need to register and pay an additional £30 fee for each course you enrol on. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online. If you do not register when you enrol, you have up until the course start date to register and pay the £30 fee. 

See more information on CATS point

Coursework is an integral part of all online courses and everyone enrolled will be expected to do coursework, but only those who have registered for credit will be awarded CATS points for completing work at the required standard. If you are enrolled on the Certificate of Higher Education, you need to indicate this on the enrolment form but there is no additional registration fee. 

 

Digital credentials

All students who pass their final assignment, whether registered for credit or not, will be eligible for a digital Certificate of Completion. Upon successful completion, you will receive a link to download a University of Oxford digital certificate. Information on how to access this digital certificate will be emailed to you after the end of the course. The certificate will show your name, the course title and the dates of the course you attended. You will be able to download your certificate or share it on social media if you choose to do so. 

Please note that assignments are not graded but are marked either pass or fail. 

Fees

Description Costs
Course Fee £285.00
Take this course for CATS points £30.00

Funding

If you are in receipt of a UK state benefit, you are a full-time student in the UK or a student on a low income, you may be eligible for a reduction of 50% of tuition fees. Please see the below link for full details:

Concessionary fees for short courses

Tutor

Dr Tom Crawford

Tom is the Public Engagement Lead at the Department and is also a Fellow by Special Election at St Edmund Hall where he teaches mathematics to the first and second year undergraduate students.

Alongside his teaching commitments, Tom runs an award-winning outreach programme through his website ’Tom Rocks Maths’ which hosts videos, podcasts, puzzles and articles that aim to make maths entertaining and understandable to all. Tom works with several partners including the BBC and the Numberphile YouTube channel – the largest maths channel on the platform with over pi-million subscribers. With over 20 million YouTube views, 2 TEDx talks, and guest lectures at the Royal Institution and New Scientist Live, Tom is well on his way to his goal of bringing maths to the masses.

Course aims

Course aim: Develop a deeper knowledge of Calculus through the study of Multi-Variable Functions and Partial Differential Equations. Follows OUDCE’s ‘Introduction to Calculus' course.

Course objectives: 

- Introduce the concept of mathematical modelling through the vehicle of Calculus;

- Extend student's knowledge beyond the basics of computation, to an understanding of theory and derivation of formulae;

- Develop the high-level analytical skills required of a Mathematician.

Teaching methods

The course will consist of the following:

- A weekly lecture video (50-60 minutes averaged over the 10 weeks) to cover the core concepts of each topic

- Guided reading of lecture notes, textbooks, and sample exercises

- A weekly problem set

- A 1-hour weekly group tutorial to cover the solutions to the problem set and answer any questions about the content

Learning outcomes

By the end of the course students will be expected to:

- Solve simple first and second order ODEs using the techniques of Integrating Factors, Homogenous Functions, and Separation of Variables;

- Apply the tools of Calculus to develop Mathematical Models for a variety of real-world situations, including optimisation problems;

- Demonstrate an understanding of multi-variable functions through Partial Differentiation and Taylor Series, as well as employing the techniques of Separable Solutions and Fourier Series to solve PDEs.

Assessment methods

Weekly problem sets on the lecture material will be used to determine the content of the weekly tutorials. There will also be a 'mock' exam in week 5 as a formative assessment and practice for the final exam at the end of the course. The final exam will be untimed, open-book and will cover all topics in the course. It will determine the final grade.

 

Coursework is an integral part of all weekly classes and everyone enrolled will be expected to do coursework in order to benefit fully from the course. Only those who have registered for credit will be awarded CATS points for completing work the required standard.

Students must submit a completed Declaration of Authorship form at the end of term when submitting your final piece of work. CATS points cannot be awarded without the aforementioned form - Declaration of Authorship form

Application

We will close for enrolments 7 days prior to the start date to allow us to complete the course set up. We will email you at that time (7 days before the course begins) with further information and joining instructions. As always, students will want to check spam and junk folders during this period to ensure that these emails are received.

To earn credit (CATS points) for your course you will need to register and pay an additional £30 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online.

Please use the 'Book' or 'Apply' button on this page. Alternatively, please complete an Enrolment Form (Word) or Enrolment Form (Pdf)

Level and demands

Knowledge of differentiation is required, including the limit definition of a derivative; the rules of differentiation such as the chain rule, product rule and quotient rule; as well as familiarity with vectors, their properties and manipulation. Some basic knowledge of Physics would be helpful, but is not essential.

A typical student on this course will have studied Calculus at high school (at some point in the past), and may have taken an Introductory Calculus course at College or University. This course is designed to follow on from OUDCE’s ‘Introduction to Calculus' course.

The Department's Weekly Classes are taught at FHEQ Level 4, i.e. first year undergraduate level, and you will be expected to engage in a significant amount of private study in preparation for the classes. This may take the form, for instance, of reading and analysing set texts, responding to questions or tasks, or preparing work to present in class.

Credit Accumulation and Transfer Scheme (CATS)

To earn credit (CATS points) you will need to register and pay an additional £30 fee per course. You can do this by ticking the relevant box at the bottom of the enrolment form or when enrolling online. Students who register for CATS points will receive a Record of CATS points on successful completion of their course assessment.

Students who do not register for CATS points during the enrolment process can either register for CATS points prior to the start of their course or retrospectively from the January 1st after the current full academic year has been completed. If you are enrolled on the Certificate of Higher Education you need to indicate this on the enrolment form but there is no additional registration fee.

Terms & conditions for applicants and students

Information on financial support